Polar orbitopes

We study polar orbitopes, i.e. convex hulls of orbits of a polar representation of a compact Lie group. They are given by representations of K on p, where K is a maximal compact subgroup of a real semisimple Lie group gG with Lie algebra g= k \oplus p. The face structure is studied by means of the gradient momentum map and it is shown that every face is exposed and is again a polar orbitope. Up to conjugation the faces are completely determined by the momentum polytope. There is a tight relation with parabolic subgroups: the set of extreme points of a face is the closed orbit of a parabolic subgroup of G and for any parabolic subgroup the closed orbit is of this form.